Kahan's exp quotient

Average Error: 40.1 → 0.3

Time: 8.3s

Precision: 64

Math

\[\frac{e^{x} - 1}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.900630394348508378759426973303447994113 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{5}{96}, x, \frac{1}{4}\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{5}{96}, \frac{1}{4}\right), 1\right)\\ \end{array}\]

C

double f(double x) {
        double r73800 = x;
        double r73801 = exp(r73800);
        double r73802 = 1.0;
        double r73803 = r73801 - r73802;
        double r73804 = r73803 / r73800;
        return r73804;
}

↓

double f(double x) {
        double r73805 = x;
        double r73806 = -0.00019006303943485084;
        bool r73807 = r73805 <= r73806;
        double r73808 = r73805 + r73805;
        double r73809 = exp(r73808);
        double r73810 = 1.0;
        double r73811 = r73810 * r73810;
        double r73812 = r73809 - r73811;
        double r73813 = exp(r73805);
        double r73814 = r73813 + r73810;
        double r73815 = r73812 / r73814;
        double r73816 = r73815 / r73805;
        double r73817 = 0.052083333333333336;
        double r73818 = 0.25;
        double r73819 = fma(r73817, r73805, r73818);
        double r73820 = 1.0;
        double r73821 = fma(r73805, r73819, r73820);
        double r73822 = fma(r73805, r73817, r73818);
        double r73823 = fma(r73805, r73822, r73820);
        double r73824 = r73821 * r73823;
        double r73825 = r73807 ? r73816 : r73824;
        return r73825;
}

Error

Bits error versus x

Target

Original	40.1
Target	40.5
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -0.00019006303943485084

   1. Initial program 0.1

      \[\frac{e^{x} - 1}{x}\]
   2. Using strategy rm

   3. Applied flip-- 0.1

      \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}{x}\]

   4. Simplified 0.1

      \[\leadsto \frac{\frac{\color{blue}{e^{x + x} - 1 \cdot 1}}{e^{x} + 1}}{x}\]

   if -0.00019006303943485084 < x

   1. Initial program 60.2

      \[\frac{e^{x} - 1}{x}\]

   2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]

   3. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.4

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)}}\]

   6. Simplified 0.4

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), 1\right)}} \cdot \sqrt{\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)}\]

   7. Simplified 0.4

      \[\leadsto \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), 1\right)} \cdot \color{blue}{\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), 1\right)}}\]

   8. Taylor expanded around 0 0.4

      \[\leadsto \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), 1\right)} \cdot \color{blue}{\left(\frac{5}{96} \cdot {x}^{2} + \left(\frac{1}{4} \cdot x + 1\right)\right)}\]

   9. Simplified 0.4

      \[\leadsto \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), 1\right)} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{5}{96}, \frac{1}{4}\right), 1\right)}\]

   10. Taylor expanded around 0 0.4

       \[\leadsto \color{blue}{\left(\frac{5}{96} \cdot {x}^{2} + \left(\frac{1}{4} \cdot x + 1\right)\right)} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{5}{96}, \frac{1}{4}\right), 1\right)\]

   11. Simplified 0.4

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{5}{96}, x, \frac{1}{4}\right), 1\right)} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{5}{96}, \frac{1}{4}\right), 1\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.900630394348508378759426973303447994113 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{5}{96}, x, \frac{1}{4}\right), 1\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{5}{96}, \frac{1}{4}\right), 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))